Internal-Wave-Driven Mixing: Global Geography and Budgets

Internal-Wave-Driven Mixing: Global Geography and Budgets

ERIC KUNZE

NorthWest Research Associates, Redmond, Washington

(Manuscript received 13 June 2016, in final form 6 January 2017)

ABSTRACT

Internal-wave-driven dissipation rates « and diapycnal diffusivitiesK are inferred globally using a finescale

parameterization based on vertical strain applied to;30 000 hydrographic casts. Global dissipations are 2.060.6 TW, consistent with internal wave power sources of 2.1 6 0.7 TW from tides and wind. Vertically in-

tegrated dissipation rates vary by three to four orders of magnitude with elevated values over abrupt to-

pography in the western Indian and Pacific as well as midocean slow spreading ridges, consistent with internal

tide sources. But dependence on bottom forcing is much weaker than linear wave generation theory, pointing

to horizontal dispersion by internal waves and relatively little local dissipation when forcing is strong.

Stratified turbulent bottom boundary layer thickness variability is not consistent with OGCM parameteri-

zations of tidal mixing. Average diffusivities K 5 (0.3–0.4) 3 1024 m2 s21 depend only weakly on depth,

indicating that « 5 KN2/g scales as N2 such that the bulk of the dissipation is in the pycnocline and less than

0.08-TW dissipation below 2000-m depth. Average diffusivities K approach 1024 m2 s21 in the bottom

500 meters above bottom (mab) in height above bottom coordinates with a 2000-m e-folding scale. Average

dissipation rates « are 1029W kg21 within 500 mab then diminish to background deep values of 0.15 31029W kg21 by 1000 mab. No incontrovertible support is found for high dissipation rates in Antarctic Cir-

cumpolar Currents or parametric subharmonic instability being a significant pathway to elevated dissipation

rates for semidiurnal or diurnal internal tides equatorward of 288 and 148 latitudes, respectively, althoughelevated K is found about 308 latitude in the North and South Pacific.

1. Introduction

Quantifying and understanding ocean mixing remains

one of the most challenging problems in physical

oceanography because of its spatial and temporal het-

erogeneity. Much of the turbulent mixing is concen-

trated in localized hot spots so that average mixing can

only be accurately assessed from large amounts of data

with well-distributed geographical coverage (Kunze

et al. 2006; Whalen et al. 2012; Waterhouse et al. 2014).

A wide range of features on time scales of months to

millennia, from precipitation in the western equatorial

Pacific (Jochum 2009) to the strength of the deep me-

ridional overturning circulation, equatorial upwelling,

and the Southern Hemisphere westerlies (Friedrich

et al. 2011; Melet et al. 2016), are sensitive to how dia-

pycnal mixing is parameterized in global OGCMs,

linking diapycnalmixing not just to the ocean circulation

but also biogeochemical cycles, weather, and long-term

climate. Jochum (2009) found that applying the latitude

dependence to mixing (Gregg et al. 2003) improved the

skill of OGCMs in reproducing equatorial SST and

precipitation, the spiciness of Labrador Seawater, and

the Gulf Stream path.

In the bulk of the stratified ocean interior, internal

wave breaking is the dominant source of turbulentmixing

(Munk and Wunsch 1998). The primary sources of

deep-ocean internal waves are tide/topography generation

of internal tides at ;1.0TW (Egbert and Ray 2001;

Nycander 2005); wind-forced, near-inertial waves at

0.2–1.1 TW (Alford 2001; Plueddemann and Farrar 2006;

Furuichi et al. 2008; Rimac et al. 2013); and subinertial

flow/topography generation of internal lee waves at

0.2–0.7 TW (Scott et al. 2011; Nikurashin and Ferrari 2011;

Wright et al. 2014). Thus, total internal wave power input

is 2.1 6 0.7 TW with most of the uncertainty in (i) near-

inertial wave production by winds, associated with the

temporal resolution of global wind products at high lati-

tudes and mixed-layer depth assumptions, and (ii) lee-

wave dissipation based on microstructure measurementsCorresponding author: Eric Kunze, [email protected]

Denotes content that is immediately available upon publica-

tion as open access.

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DOI: 10.1175/JPO-D-16-0141.1

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being up to an order of magnitude below predictions

(Waterman et al. 2014). This generated internal wave en-

ergy is redistributed vertically and horizontally by propa-

gation of low-mode internal waves (Ray and Mitchum

1997; Alford 2001; Zhao et al. 2016) and from large to

small scales by wave–wave and wave–mean flow in-

teractions. Away from direct forcing by wind and currents

at boundaries, turbulence production is controlled by the

rate at which energy cascades from large to small vertical

scales. Internal wave–wave interaction theory (McComas

and Müller 1981; Henyey et al. 1986; Henyey 1991) has

provided a parameterization for turbulence production

that can be expressed in terms of finescale internal wave

shear Vz and/or strain jz (Gregg 1989; Gregg and Kunze

1991; Polzin et al. 1995; Gregg et al. 2003). Comparison

with direct microstructure measurements validates the

shear-and-strain finescale parameterizations to within

factors of 2–3 (Polzin et al. 1995; Polzin et al. 2014;Whalen

et al. 2015), although care is needed in its implementation

to avoid overestimation, particularly in low-N environ-

ments where sensor noise becomes problematic andwhere

N exhibits large changes with depth.

The finescale parameterization has recently been

reviewed by Polzin et al. (2014). In contrast to direct

microstructure measurements, its inferences represent

turbulent dissipation and mixing with built-in averaging

over internal wave time and space scales. It has been

used to (i) infer that elevated dye mixing in Santa

Monica Basin (Ledwell and Watson 1991) was being

driven by an energetic internal wave field on the slopes

(Gregg and Kunze 1991); (ii) predict elevated turbulent

mixing above seamount summits (Kunze et al. 1992)

before microstructure confirmation (Lueck and Mudge

1997; Toole et al. 1997; Kunze andToole 1997); (iii) infer

weak mixing over smooth bottom topography and ele-

vated mixing over rough topography (D’Asaro and

Morison 1992; Wijesekera et al. 1993; Kunze and

Sanford 1996; Mauritzen et al. 2002; Walter et al. 2005;

Kunze et al. 2006; Stöber et al. 2008; MacKinnon et al.

2008), consistent with deep direct microstructure mea-

surements (Toole et al. 1994; Polzin et al. 1997);

(iv) identify sites whereAntarctic Circumpolar Currents

interact with bottom topography to generate elevated

turbulence northeast of the Kerguelan Plateau (Polzin

and Firing 1997; Kunze et al. 2006) and inDrake Passage

(Naveira Garabato et al. 2004; Wu et al. 2011; Damerell

et al. 2012), subsequently verified with direct micro-

structure measurements (St. Laurent et al. 2012; Sheen

et al. 2013; Waterman et al. 2013); (v) contribute to the

argument that parametric subharmonic instability (PSI)

may transfer energy from low-mode internal tides to

high-wavenumber near-inertial shear of half the fre-

quency (Hibiya et al. 2006) and thence to turbulence

immediately equatorward of 288 (MacKinnon and

Winters 2005; Carter and Gregg 2006; Simmons 2008;

MacKinnon et al. 2013a,b; Sun and Pinkel 2013);

(vi) determine there is little seasonal variability in upper-

ocean mixing except under fall–winter storm tracks

(308–408) (Whalen et al. 2015); and (vii) assess the role of

turbulent diapycnal mixing in the meridional over-

turning circulation and large-scale property budgets in

the Indian Ocean (Huussen et al. 2012).

In this paper, a global assessment of deep-ocean,

internal-wave-driven turbulent dissipation rates « and

diapycnal diffusivities K will be inferred by applying a

parameterization based on finescale internal-wave

strain jz (Gregg and Kunze 1991; Wijesekera et al. 1993;

Polzin et al. 1995; Gregg et al. 2003) to ;30 000 CTD

profiles. Data are absent from the Arctic, Weddell, and

Ross Seas and are limited poleward of 608S in the

Southern Ocean but otherwise are well-distributed with

latitude and longitude in all the major ocean basins.

Strain-based inferenceof internal-wave-driven turbulence

dissipation rates « and diffusivitiesK has seen widespread

use (Mauritzen et al. 2002; Sloyan 2005; Lauderdale et al.

2008; Wu et al. 2011; Whalen et al. 2012; 2015; Damerell

et al. 2012). The analysis here expands on previous shear-

and-strain estimates based on ;3500 full-depth LADCP/

CTD profiles (Kunze et al. 2006) with an order of mag-

nitude more data to provide much more comprehensive

global geographical coverage, expanding the scope to the

South Atlantic and improving sampling in the North At-

lantic, western Pacific, and Southern Oceans. The data

and methods are detailed in section 2, global maps and

sections are discussed in section 3, averages and budgets

are in section 4, zonally averaged structure is in section 5,

an evaluation of OGCM tidal mixing parameterizations is

in section 6, and finally the conclusions and discussion are

found in section 7. In a companion paper (Kunze 2017,

manuscript submitted to J. Phys. Oceanogr.), the in-

ferred diapycnal diffusivities K and dissipation rates

« are used to compute the interior internal-wave-driven

diabatic meridional overturning circulation and com-

pare it with diapycnal transports driven by near-bottom

buoyancy-flux divergence.

2. Data and methods

CTD profiles from WOCE/CLIVAR hydrographic

sections (https://cchdo.ucsd.edu/) were employed in this

analysis. These include about 73 sections from the In-

dian, 422 from the Pacific [although over 200 of these are

from theHawaii Ocean Timeseries (HOT) site], and 146

from the Atlantic with roughly 6700, 13 000, and 10 500

usable casts, respectively; casts shorter than 300m or

with resolution coarser than 2m are excluded. No data

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are included from the Arctic, Weddell, or Ross Seas and

are sparse south of 608S in the SouthernOcean. Roughly

10% of the retained profiles span less than 25% of the

water column. The T, S, and p files were downloaded

and converted to a uniform format. Density variables sp,

su, s3, and gn and stratificationN2 were then computed.

Internal-wave-driven turbulent dissipation rates « and

diapycnal diffusivities K are inferred from a finescale pa-

rameterization based on internal wave–wave interaction

theory (McComasandMüller 1981;Henyey et al. 1986) that

was first tested using 10-m vertical shear (Gregg 1989), then

modified to estimate variances spectrally in vertical wave-

number space and incorporate internal wave strain as both

(i) an independentmeans of inferring internalwave spectral

levels (Gregg and Kunze 1991; Wijesekera et al. 1993) and

(ii) to account for deviations of the internal wave aspect

ratio or frequency content from the Garrett–Munk (GM)

model (Polzin et al. 1995). The strain-based form of the

parameterization used here is

K5K0

hj2zi2

GMhj2zi2h(R

v)j

�N

f

�(1)

(e.g., Kunze et al. 2006), where K0 5 0.05 3 1024m2 s21

for a mixing efficiency g 5 0.2, hjz2i is the strain variance

with strain estimated as jz 5 (N2 2 N2fit)/N

2fit following

Polzin et al. (1995), and N2fit represents a quadratic fit to

half-overlapping, 256-m profile segments; alternative fit-

ting procedures were attempted including to log(N2)

rather than N2 and different functional forms, but these

were found to be biased compared to the simpler fit-

ting scheme. The normalizing GM model strain variance

GMhjz2i is computed over the same wavenumber band as

the observed strain. The GM75 model vertical wave-

number kz spectrum for strain is given by

SGM

[jz](k

z)5

pE0b

2

k2zj*

(kz1 k

z*)2

(2)

(Cairns and Williams 1976; Gregg and Kunze 1991),

where the canonical nondimensional spectral energy

level E0 5 6.3 3 1025, stratification length scale b 51300m, peak mode number j* 5 3, and corresponding

vertical wavenumber kz* 5 (pj*/b)(N/N0). Integrated to

10-m vertical wavelengths, the GM75 strain variance in

(2) is 0.24. The dependence on shear/strain variance

ratio Rv 5 hVz2i/(hN2ihjz2i) is

h(Rv)5

1

6ffiffiffi2

p Rv(R

v1 1)ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Rv2 1

p (3)

(Fig. 1a), which provides a crude measure of the wave

field’s aspect ratio (or frequency) content for internal

waves dominated by lower frequencies (Henyey 1991;

Kunze et al. 1990; Polzin et al. 1995). The dependence on

the ratio of buoyancy to Coriolis frequencies N/f is

j

�N

f

�5

fArccosh(N/f )

f30Arccosh(N

0/f30), (4)

where f30 5 f(308) 5 7.3 3 1025 rad s21 and N0 5 5.2 31023 rad s21. The latitude dependence [(4)] vanishes at

the equator because f goes to zero more rapidly than

Arccosh(N/f) goes to infinity; the appendix argues that

internal waves at the equator will be equatorially trap-

ped meridional modes with minimum frequencies set by

off-equatorial Coriolis frequencies at turning latitudes,

but it is not known how this will impact the cascade. The

FIG. 1. (a) Correction h(Rv) [(3)] for the strain-only turbulence

parameterization [(1)] as a function of shear/strain variance ratio

Rv. (b) Probability distribution function of Rv based on the

LADCP/CTD analysis of Kunze et al. (2006) with vertical lines

denoting the GM Rv 5 3 as well as the measured mean (10.6) and

mode (6) in the data. Almost identical distributions are foundwhen

the data are split into nine buoyancy frequencyN and nine latitude

bins, indicating that this distribution is a robust feature of the ocean

internal wave field. Means (modes) decrease for larger strain var-

iances that will contribute more to mixing, dropping to 8 (4) for

hjz2i/GM exceeding 1.

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dissipation rate « 5 KhN2i/g (Osborn 1980), where

mixing efficiency g 5 0.2 is assumed (Oakey 1982;

Itsweire et al. 1986;Moum 1996; St. Laurent and Schmitt

1999) following standard practice in the ocean micro-

structure observational community.

For the GM model, the shear/strain variance ratio

GMRv 5 3. But the ocean’s average Rv appears to be

higher (Fig. 1b), signifying that the ocean is more in-

ertial than the GMmodel on average. For measuredRv

data (Kunze et al. 2006), the distribution has mean 10.6

and mode 6.1, independent of buoyancy frequency N

and latitude; 80% of the data had Rv below 16. The

distribution’s mean and mode diminish for the larger

strain variances that will dominate internal-wave-

driven mixing (not shown). For shear/strain ratios

greater than 3, h(Rv) is an increasing function of Rv;

Rv 5 7 will be used here, which produces dissipation

rates « and diffusivities K a factor of 3 larger than for

Rv 5 3 and a factor of 3 smaller than for Rv 5 10

(Fig. 1a), so that factor of 3 uncertainties are expected;

Kunze et al. (2006) reported maximum factor of 2 dif-

ferences between shear-and-strain and strain-only dif-

fusivities for Rv 5 7.

Finescale parameterization (1) only accounts for

weakly nonlinear internal-wave-driven turbulence. It will

fail in environments where a weakly nonlinear wave-

number cascade is not expected either because of (i) lack

of bandwidth such as on continental shelves (MacKinnon

andGregg 2003; Carter et al. 2005), (ii) short-circuiting of

the cascade because of near-critical bottom reflection

(Carter and Gregg 2002; Nash et al. 2004; Kunze et al.

2012), or (iii) direct boundary forcing of turbulence (e.g.,

Klymak et al. 2008, 2010). It does not account for mixing

due to hydraulically controlled flow (Ferron et al. 1998)

or breaking solitary waves (MacKinnon andGregg 2003).

While these regions occupy a small fraction of the ocean,

theymay be important. For example, density overturns of

O(100) m [e.g., in Luzon Strait (Alford et al. 2011), Sa-

moan Passage (Alford et al. 2013), and Romanche

Fracture Zone (Ferron et al. 1998)] imply local diffusiv-

ities 104–105 times the background and so they need only

occupy 0.01%–0.1% of the ocean to produce basin-

averaged diffusivities of 1024m2s21. The shear-and-strain

parameterization overestimates turbulent dissipation rates

on the flanks of the Florida Strait (Winkel et al. 2002) and

overlying regions where lee-wave generation is expected

(Waterman et al. 2014).

Strain variance for (1) is estimated spectrally from

strain jz 5 (N2 2 N2fit)/N

2fit for half-overlapping, 256-m

profile segments starting at the bottomup to the depth of

the highest N2 in the upper 150m of the water column

(to exclude the surface mixed layer). This yields roughly

500 000 usable estimates. Strain variances are computed

by integrating the strain spectra S[jz](kz) from the

lowest resolved vertical wavenumber (lz5 256m) to the

wavenumber where variance exceeds a threshold value

of 0.05, which, for a GM-level spectrum, corresponds to

lz 5 50m, in part to avoid contamination by ship heave

near 10-m wavelengths (Polzin et al. 2014). By stopping

the integration short of the rolloff wavenumber, it is

assumed that the strain spectrum is flat like the GM

model; redder (more negative spectral slope) spectra, as

typically found here, will result in overestimation of the

strain variance by at most a factor of 1.3, while bluer

(more positive spectral slope) spectra will result in un-

derestimation. The 256-m profile segment length is a

compromise between resolution in depth and strain

variance (diffusivity). Strain variance is computed below

the rolloffwavenumberkc, whichbehaves as (0.2pm)EGM/E

(Fritts 1984; Gargett 1990; D’Asaro and Lien 2000).

With diffusivities K(EGM) ; 1025m2 s21 and K ; E2,

256-m segment lengths can resolve strain variance ra-

tios hjz2i/GMhjz2i less than 10 and diffusivitiesK less than

O(1023) m2 s21.

Strain is assumed to be dominated by finescale in-

ternal waves. While contamination by finescale geo-

strophic motions (Pinkel 2014), thermohaline staircases

(Gregg 1989), and interleaving cannot be ruled out on

dynamical grounds, in earlier high-resolution profiler

and CTD analyses, the only contamination signals that

stood out were sharp pycnoclines at low latitudes (Polzin

et al. 1995;Mauritzen et al. 2002; Kunze et al. 2006), with

contamination bywater-mass intrusions and geostrophic

motions appearing to be confined to wavelengths less

than 10m (Polzin et al. 2003) and greater than 200m

(Kunze et al. 2006). Therefore, the contamination is

largely filtered out here by the chosen 50–256-m band of

integration. Pinkel (2014) reports subinertial strain

confined to near the base of the mixed layer. Double-

diffusive interleaving may contribute where there is

strong water-mass variability on isopycnals (S. Merrifield

2016, personal communication). Double diffusion

tends to produce thermohaline staircases in only a few

known locales of weak internal-wave-driven mixing

(Gregg and Sanford 1987; Kunze 2003), which are of

little interest here, or at lateral water-mass boundaries.

Thermohaline staircases escape the spectral filter at

1200–1800-m depth beneath the Mediterranean salt

tongue between 308 and 408N in the eastern North

Atlantic and are also expected east of Barbados, in the

Tyrrhenian Sea, and under the Red Sea outflow in the

Arabian Sea (Schmitt 2003).

To avoid contamination by sharp pycnoclines, the

shallowest two segments (corresponding to the upper

380m of the water column) are omitted from anal-

ysis because they often, and unpredictably, exhibit

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unrealistically high strain variances. This problem has

previously been recognized and dealt with in a similar

manner by Mauritzen et al. (2002), Kunze et al. (2006),

andWhalen et al. (2012).Whalen et al. (2015) compared

Argo float strain-based diffusivities K to average mi-

crostructure K profiles at six sites and found that 81%

agreed to within a factor of 2 and 96% agreed to within a

factor of 3 below 250-mdepth.Amore conservative 380-m

depth was chosen here because of contamination by

very deep mixed layers at high latitudes. Qualitative

evidence that this is sufficient can be seen in Figs. 4, 6,

and 9 (shown below), where the shallowest plotted dif-

fusivities and strain variances just below 380m are

similar to those at greater depths. Profile segments were

also excluded if their average buoyancy frequency hNifell below the noise threshold 33 1024 rad s21, as these

are dominated by digitization noise (Whalen et al. 2015).

With the expectation thatN. 2f is a minimal frequency

bandwidth to allow internal wave–wave interactions,

segments with hNi less than 2f were also excluded; these

largely overlap with the hNi noise threshold, repre-

senting 10% of the data, 30% within 380 meters above

bottom (mab), and 17% within 1000 mab in abyssal

basins. Very low stratification is found (i) throughout

much of the water column at high latitudes in the

Southern Ocean, particularly the eastern Atlantic sec-

tor; (ii) at middepth around southern Greenland; and

(iii) in abyssal basins in the Caribbean, Angola Basin,

and North Pacific. These will have small diapycnal

buoyancy fluxes hw0b0i 5 2KhN2i, where w0 and b0 areturbulent vertical velocity and buoyancy fluctuations,

because their stratification is weak, so their omission has

little impact. While internal waves can exist for N , f

such that topographic generation of internal waves is

still possible in very weakly stratified bottom boundary

layers, these will be unable to propagate into regions

where N exceeds f and so will be confined near the

bottom. How this weak stratification might impact in-

ternal tide and lee-wave generation, or near-bottom

turbulence, has not been investigated to the author’s

knowledge.

3. Geography

Maps of depth-integrated dissipation rateЫ5 r0

Ыdz1

from 380-m depth to the bottom (Fig. 2) show elevated

values associated with abrupt topography and slow

spreading ridges, particularly (i) in the western In-

dian over the Southwest Indian Ridge (308–408S,408–608E), the Seychelles and Mascarene Ridge east of

Madagascar (28–208S, 508–608E), in the western Arabian

Sea over the Owen Fracture Zone (108N, 608E) and

Carlsberg Ridge, and in the wake of the Kerguelan

Plateau (508–608S, 708–808E; Polzin and Firing 1997);

(ii) in the western Pacific over abrupt ridge/trench to-

pography associated with subduction; (iii) in the central

Pacific associated with island archipelagos such as the

Hawaiian Island chain (208–308N, 1508–1808W) and

Tuamotu Archipelago (208–308S, 1308–1608W); and

(iv) in theAtlantic over continental slopes such as in theBay

of Biscay (458–508N, 08–108W), the Flemish Cap (408–508N,358–408W), and Mid-Atlantic Ridge. Low values are

found over smooth abyssal basins such as (i) the

central Arabian Sea (108–158N, 558E) and Bay of Bengal

(108–208N, 908E) and south Indian Basin south-

southeast of Sri Lanka (08–208S, 808E) in the Indian

Ocean, (ii) in the subpolar North Pacific and eastern

Pacific Ocean, (iii) in the Angola Basin in the eastern

South Atlantic (08–208S, 08–208E), and (iv) in the

Southern Ocean south of ;588S, where N is low

throughout the water column. Overall, high vertically

integrated dissipation rates are consistent with predicted

sites of high internal tide generation (e.g., Egbert and

Ray 2001; Simmons et al. 2004; Nycander 2005). Low

values in the Southern Ocean do not support lee waves

being a major dissipative conduit for the Antarctic

Circumpolar Currents (Nikurashin and Ferrari 2011;

Scott et al. 2011), consistent with microstructure measure-

ments at Kerguelen Plateau and Drake Passage finding

dissipation rates up to an order of magnitude below lee-

wave generation predictions (Waterman et al. 2014).

The WOCE/CLIVAR hydrography is not well suited

for examining the upper ocean’s response to storm

forcing because of its temporal intermittency. Argo

profiling floats provide better temporal sampling of the

seasonal cycle of upper-ocean mixing, which appears to

be confined to 308–408 latitude (Whalen et al. 2012).

Binning vertically integrated dissipation ratesЫ by

longitude shows that elevated values in the western In-

dian and western to central Pacific are related to to-

pography (Fig. 3), not tidal flows, which is more uniform.

This interpretation augments that of Hibiya et al. (1999),

who predicted western intensification of turbulence in

the North Pacific because of the hot spot of near-inertial

wave generation near 408N and west of the date line

(Alford 2001). On average, most of the dissipation oc-

curs in the pycnocline rather than near the bottom

(Fig. 3a), in contrast to OCGM tidal mixing parame-

terizations (Jayne and St. Laurent 2001; Decloedt and

Luther 2010).

1 A factor of g21 5 5 coding error in Kunze et al. (2006) that

underestimated integrated dissipation ratesЫ in their Figs. 5–12

has been corrected here. Their conclusion that there is insufficient

dissipation to account for internal wave power sources is invalid.

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FIG. 2. Maps of the vertically integrated dissipation rateЫ5

Ðr« dz in mWm22 for the full

water column excluding the upper 380m. Low values are found in the Southern Ocean, the

Indian’s Bay of Bengal, and South Atlantic’s Angola Basin. High values are associated with

abrupt topography in the western Indian, western and central Pacific, and over midocean

ridges. Because of the N2 scaling of «, dissipation rates integrated over the pycnocline

(,2000m) show similar patterns and magnitudes.

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Repeat sections of inferred diapycnal diffusivity K

illustrate that elevated variability related to topography

is reproducible. While elevated diffusivities K above

weak rough topography are sometimes confined to

within 500mab over stronger and more extensive to-

pography, it often extends throughout the entire water

column (Fig. 4), consistent with Fig. 3a and in contrast

with the fixed decay scale of 500 m implemented in

OGCM tidal mixing parameterizations (Simmons et al.

2004; Saenko andMerryfield 2005; Jayne 2009; Friedrich

et al. 2011) based on microstructure measurements on

theMid-Atlantic Ridge bounding the eastern side of the

Brazil Basin (St. Laurent et al. 2001). This is more

clearly illustrated in the joint probability density

function of diffusivity above 2000-m depthKpycno versus

below 2000-m depth Kdeep (Fig. 5), which reveals a

correlation between diffusivities in these two depth

ranges with Kpycno ; Kdeep/2. Since « 5 KN2/g and N2

exhibit more variability than K, most of the dissipation

will be in the high stratification of the pycnocline.

Figure 4 also illustrates some of the variety of bottom

geometries that can contribute to elevated strain

variance.

Equatorial crossings consistently show elevated strain

variance within628 of the equator (Fig. 6). There is littlecorresponding signal in diapycnal diffusivityK because a

reduced cascade rate as f / 0 in (4) allows more vari-

ance to accumulate at lower wavenumbers for the same

FIG. 3. Vertically integrated dissipation ratesЫ (a) reveal that most of the dissipation

occurs in the pycnocline (red). Peaks in the western Indian and western Pacific (left two gray

bars) appear to bemost correlated with 30 km3 30 km topographic roughness h2 (b), whereas

rms tidal velocities (c) are more uniform. Elevated dissipation rates and topography over

2858–3008E (608–758W) are associated with both Drake Passage and the Caribbean. TheseЫ

exclude the upper 380m.

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dissipation rate (Gregg et al. 2003). This contrasts with

the stripes of elevated diffusivity K or integrated dissi-

pation rateЫ reported flanking the equator by Whalen

et al. (2012); the absence of this signal here appears to be

due to the hydrography casts missing frequent but in-

termittent mixing bursts associated with negative La

Niña and neutral ENSO conditions (C. B. Whalen 2016,

personal communication).

FIG. 4. Repeat sections of inferred turbulent diapycnal diffusivityK show the influence of rough topography including (a05) crossing of

the Mid-Atlantic Ridge near 258N, (p14) crossing of the Aleutian Island Ridge, (p06) crossing of the Colville Ridge and Louisville

Seamount Chain near 338S in the subtropical South Pacific, (i05) crossing the ridges in the western Indian near 348S, and (p06) crossing

between eastern Australia and north of New Zealand near 308S. While elevated diffusivities K over rough topography are sometimes

confined near the bottom, they often extend through the entire water column, which will produce particularly strong dissipation rates in

the pycnocline where N is high.

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4. Averages and budgets

The global-integrated dissipation rate r0ÐÐ Ð

«dV,

computed as r0Ð h«(z)i dz times the ocean area,

where , . is the average of all the profiles, is 1.5 60.4TW (4.3 6 1.0mWm22 per unit area) below 380-m

depth. Averages per unit area are largest in the North

Pacific (5.3mWm22) and smallest in the South Atlantic

(2.3mWm22). The above value is missing the contri-

bution above 380-m depth, which is potentially signifi-

cant in light of theN2 scaling of «, so this fraction is now

estimated.Assuming a conservative, that is, low average

pycnocline diffusivity K 5 0.1 3 1024m2 s21 (Gregg

1989; Waterhouse et al. 2014) and « 5 KN2/g above

380m implies an additional 0.5 6 0.2 TW in the upper

380m, likely an underestimate because shipboard sam-

pling is biased toward fair weather so will miss some of

the wind-forced contribution in the 308–408 latitude band(Whalen et al. 2015). The total inferred dissipation of

2.06 0.6 TW is then consistent with the sum of internal

wave power inputs of 1.0–1.2TW from the tide (Egbert

and Ray 2001; Nycander 2005), 0.2–1.1TW from wind

(Alford 2001; Plueddemann and Farrar 2006; Furuichi

et al. 2008; Rimac et al. 2013), and 0.2–0.7 TW from lee-

wave generation (Scott et al. 2011; Nikurashin and

Ferrari 2011; Melet et al. 2014; Wright et al. 2014) and

thus is sufficient to close the internal wave energy bud-

get within the present large uncertainties for both

sources and sinks of internal waves.

Average profiles are similar in all ocean basins so

only global averages are shown (Fig. 7). Average GM-

normalized strain variances hjz2i/GM ; 2, almost in-

dependent of depth z, but increase to;3 at the bottom in

height above bottom coordinates h. Dissipation rates

« exhibit the most variability with respect to depth z and

buoyancy B ’ 2(ggn/r0), where gn is neutral density,

while GM-normalized strain variance and diapycnal dif-

fusivity K vary the most with respect to height above

bottom h. Dissipation rates « decrease with depth z.

They have a minimum of 0.2 3 1029Wkg21 between

1000 and 3000 mab and increase to a maximum of

1029Wkg21 within 500-mab of the bottom. But much of

this increase is contributed by the pycnocline. Aver-

aging only waters below 2000-m depth, the average dis-

sipation rate is 0.33 1029Wkg21 at the bottom and 0.131029Wkg21 above 1000 mab in height above bottom

coordinates.

Average diffusivities K are (0.3–0.4) 3 1024m2 s21

with little dependence on depth z but increase toward

1024m2 s21 at the bottom in height above bottom co-

ordinates h with an e-folding scale of ;2000m. This dif-

ference arises because the bottom is not always at the same

depth. Diffusivities K increase from 0.3 3 1024m2 s21 for

buoyancy B . 20.27ms22 to K5 0.7 3 1024m2 s21 at

lower buoyancy (higher density), showing weaker depen-

dence than the Lumpkin and Speer (2003) inverse esti-

mates, though lying within the latter’s uncertainties except

in 20.267 . B . 20.269ms22 (gn 5 28.0–28.2). While

average diffusivity profiles here are lower than the 1024m2s21

reported below 1000-m depth from 17 microstructure sites

in Waterhouse et al. (2014), comparison of finescale pa-

rameterization [(1)] inferences in the vicinity of these 17

sites were consistent with microstructure averages. Most of

the sites considered in Waterhouse et al. have predicted

sources larger than depth-integrated dissipation rates, im-

plying that the sites were mostly located in net internal

wave sources rather than net sinks. This suggests that (i) the

strain parameterization is reasonable on average and (ii)

microstructure sampling has been biased toward regions of

stronger forcing, which is consistent withWaterhouse et al.

reporting that internal wave sources exceeded sinks atmost

microstructure sites, which have undersampled regions of

low tidal power (their Fig. 5b).

The strain-based average diffusivities K are a factor

of 2–3 higher than shear-and-strain-based values in

Kunze et al. (2006) above 3000-m depth but comparable

below 3000-m depth and comparable to their strain-based

estimates. They are higher for heights above bottom h

below 1000 mab; Kunze et al. reported average dissipation

rates « increasingmonotonicallywith height above bottom.

Cumulative dissipation rates, substituting for the

upper 380m as above, are concentrated in the upper

FIG. 5. Probability distribution function of diapycnal diffusivity

K in the pycnocline (,2000-m depth) and in the deep (.2000-m

depth), illustrating that elevated pycnocline diffusivities are cor-

related with deeper diffusivities but weaker by a factor of 2 (lower

thin dotted line).

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pycnocline (Fig. 8), with 80% of the dissipation below

380-m depth contributed above 1000m. Only 0.08TW

dissipates below 2000-m depth, suggesting very little

local mixing in the abyss. Roughly 20% (30%) of the

dissipation is found below h, 500 mab (1000 mab), that

is, 0.4 TW (0.6 TW). Again, these differences reflect that

the bottom is not always at the same depth.

5. Average zonal and meridional structure

Zonally averaged depth–latitude sections reveal small

differences between the three major oceans (Fig. 9). All

the oceans show a pool of elevated dissipation rates

« shallower than 2000-m depth for latitudes equatorward

of 508–608 (Fig. 9c), coinciding with the higher stratifica-

tion in the main pycnocline (Fig. 9b). This is not reflected

in the diffusivity K, which is vertically more uniform

(Fig. 9d), consistent with average dissipation rates

« scaling as N2 (Gregg and Sanford 1988). Weaker dif-

fusivities straddling the equator reflect the N/f scaling in

(4) but may be biased low (Whalen et al. 2012; Thurnherr

et al. 2015) because the rich equatorial wave field is

outside the scope of the internal gravity wave–wave in-

teraction theory behind the finescale parameterization.

Indian and Atlantic diffusivities K exceed 1024m2 s21 at

all depths for latitudes poleward of 408–508, associatedwith weaker high-latitude stratification, while the Pacific

diffusivities are more uniformly weak at subpolar

FIG. 6. Repeat sections illustrating equatorial crossings including (i04) in the Indian Ocean

along 808E and (p18) south of Baja California in the eastern Pacific. Strain variance is elevated

near the equator because the cascade proceeds more slowly, as represented by the j(N/f ) term

in (1). TheN/f scaling [(4)] compensates for the excess strain near the equator formore uniform

diffusivitiesK that are forced to zero at the equator as f/ 0. In the Pacific section, there is also

elevated strain variance associated with rough topography at 58S and 108N.

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latitudes and its stratification is stronger. An alternative

explanation is that this latitude band is associated with

internal lee-wave generation by Antarctic Circumpolar

Currents interacting with topography, where the finescale

parameterization overestimates turbulence dissipation

rates « by as much as an order of magnitude (Waterman

et al. 2014). In the Indian and Atlantic, diffusivities seem

to weaken slightly south of 608S, while they becomemore

elevated north of 608N in the Atlantic.

Zonally averaged vertically integrated dissipation

ratesЫ (Fig. 10a) appear to correlate with topographic

roughness h2 (Fig. 10b), while tidal flows are more

FIG. 7. Global average profiles of, from left to right, the number of data points n, buoyancy frequencyN, GM-normalized strain variance

hjz2i/GM, diapycnal diffusivity K, and dissipation rate « as functions of (top) depth z, (middle) height above bottom h, and (bottom)

buoyancy B5 0:32 (ggn)/r0 with neutral density gn indicated along the rightmost axis. Values are not plotted for n, 300 and are plotted

gray for n, 3000. Dotted curves in height above bottom coordinates exclude data shallower than 2000m. Normalized strain variance and

diapycnal diffusivityK are nearly independent of depth (top row) at 1.5–2 and (0.3–0.4)3 1024 m2 s21, respectively, butK decreases from

1024 m2 s21 at the bottom to 0.153 1024 m2 s21 above 5000mab in height above bottom coordinates h (middle row)with a 2000-m e-folding

scale, and the dissipation rate « is elevated below 1000mab because of both increasingK as h/ 0 and elevatedN below 1000 mab; much of

the elevated N and « BBL is contributed by shallow water less than 2000m deep. Strain-inferred diffusivities exhibit less variability with

buoyancyB (bottom row) than the Lumpkin and Speer (2003) inverse estimates though agreeing within their uncertainties except over B520.267 to 20.269m s22 (gn 5 28.0–28.2).

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uniform (Fig. 10c). In the Southern Hemisphere, a peak

inЫ poleward of 308S is not consistent with the pre-

dictions of parametric subharmonic instability enhanc-

ing the cascade of low-mode internal tide energy

equatorward of 288 and 148 (MacKinnon and Winters

2005; Hibiya et al. 2006; Alford et al. 2007; MacKinnon

et al. 2013a,b). A signature of elevatedЫ equatorward

of 308N in the Northern Hemisphere may be tied to ei-

ther parametric subharmonic instability or elevated to-

pographic roughness over 108–228N (Fig. 10b) latitude in

both the Pacific and Atlantic.

Meridionally averaged depth–longitude sections show

widespread elevated diffusivities poleward of 508 in the

Atlantic and Indian (Fig. 11) and a slight tendency to-

ward higher values in the upper ocean near western

boundaries. In the 508–708S latitude bin, the longitude

dependence resembles that of wind forcing (Kilbourne

2015) with elevated values of O(1024) m2 s21 in the In-

dian and Atlantic sectors of the Southern Ocean but

weak mixing O(1025) m2 s21 in the eastern Pacific sec-

tor. However, this pattern is also seen in higher strati-

fication N2 in the eastern Pacific sector compared to the

Indian andAtlantic (Fig. 11), and, as alreadymentioned,

this is the latitude band where the finescale parameter-

ization overestimates turbulent dissipation rates in

Antarctic Circumpolar Currents (Waterman et al. 2014).

6. Tidal mixing parameterizations

The last decade has seen the development (Jayne and

St. Laurent 2001; Polzin 2004; Decloedt and Luther

2010) and implementation (St. Laurent et al. 2002;

Simmons et al. 2004; Saenko andMerryfield 2005; Jayne

2009; Friedrich et al. 2011; Melet et al. 2013, 2014) of

subgrid-scale parameterizations for local tidally driven

mixing in OGCMs. In general, the dissipation rate « can

be expressed as

«5qE(x, y)F(z) (5)

(Jayne and St. Laurent 2001), whereE(x, y) is the laterally

variable bottom forcing, F(z) is the vertical structure, and

q is the fraction lost to turbulent dissipation locally in the

overlying water column. OGCM implementations of (5)

have assumed constant q5 0.3 and a constant decay scale

of 500m in F(z). As already shown (Figs. 4–5), turbulent

bottom boundary layer thicknesses are extremely vari-

able, often extending through the entire water column.

This supports a more dynamically variable turbulent

bottom boundary layer thickness, such as Polzin (2004) as

implemented in Melet et al. (2013), or Olbers and Eden

(2013). Most of the dissipation occurs in the pycnocline

(Figs. 3, 7, 8, 9, 10). This suggests that bottom-generated

internal tides freely propagate up through the water col-

umn, largely dissipating in the upper ocean where higher

stratification amplifies the nonlinear cascade.

Likewise, it is known that q is much lower over steep

isolated topography (Althaus et al. 2003; Klymak et al.

2006) than over the abyssal hills’ topography character-

izing slow midocean spreading ridges (St. Laurent and

Garrett 2002). Here, we compare vertically integrated

dissipation ratesЫwith topographic forcing predictions

to reiterate that q is not constant but appears to decrease

with increasing forcing. In Fig. 12, the vertically in-

tegrated dissipation ratesЫ (Fig. 2) are binned with (i)

linear internal tide power input (Bell 1975)

E(x, y)5NU2hhx5 kNU2h2 , (6)

(ii) linear internal lee-wave generation theory (Bell 1975)

E(x, y)5N2Uh2 , (7)

and (iii) topographic roughness (height variance) h2,

whereN is the bottom buoyancy frequency;U is the rms

barotropic tidal velocity from TPXO.3 (Egbert and Ray

FIG. 8. Global cumulative dissipation ratesЫ as a function of

(top) depth z and (bottom) height above bottom h illustrate that

50% (80%) of the dissipation occurs above 500-m (700m) depth.

Dissipation rates are not accumulated above 400-m depth. Only

20% of the dissipation is found below h 5 500 mab.

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2001); h2 is the topographic roughness (height variance)

on length scales less than that of the mode-one internal

tide, which here is taken to be the variance in 30km 330km domains in the Smith and Sandwell (1997) global

bathymetric database (10km 3 10km domains yielded

similar dependences); and k is a characteristic horizontal

wavenumber that is taken as a free parameter to best

match rms global surface tide elevation (Simmons et al.

2004). These linear theories are applicable for weak to-

pography (topographic height hmuch less than the water

depthH and topographic slope smuch less than the wave

ray slope k/m), which is valid at the semidiurnal frequency

for 97% of the bottom based on Smith and Sandwell

(1997) bottom slopes (though only 75% above 1500-m

water depth). At 1.0–1.2TW (Egbert and Ray 2001;

Nycander 2005), internal tide generation [(6)] is thought

to dominate (Waterhouse et al. 2014) over the less certain

wind-generated inertial waves power input of 0.2–1.1TW

(Alford 2001; Plueddemann and Farrar 2006; Furuichi

et al. 2008; Rimac et al. 2013) and internal lee-wave gen-

eration [(7)] of 0.2–0.7TW (Scott et al. 2011; Nikurashin

and Ferrari 2011; Wright et al. 2014; Waterman et al.

2014), but neither these other sources nor remote tidal

dissipation are separable in our estimates.

Observed dependence forЫ on topographic forcing

(Fig. 12) is much weaker than one to one and similar to

the h1/2 dependence reported by Kunze et al. (2006).

Since saturation has been discounted above, we in-

terpret this as signifying that bottom-forced internal

waves are not all locally dissipated (q , 1), and hori-

zontal radiation redistributes a significant fraction of

the forcing before dissipation and mixing; that is, most

internal-wave-driven mixing is remote from sources.

This is consistent with Waterhouse et al. (2014), who

reported that most of the 17 microstructure measure-

ment sites they considered had excess forcing compared

to dissipation. The weak dependence ofЫ on forcing in

Fig. 12 points to q decreasing with increasing forcing, but

there is order-of-magnitude scatter, suggesting unre-

solved physics.

While Figs. 4 and 12 point to problems with choosing

constant local dissipation fractions q and decay scales as

in existing tidal mixing parameterizations, it would be

premature to suggest better scalings. Order of magni-

tude scatter is evident in the raw scatterplots (Fig. 12)

that may be related to topographic details lost in the

coarse Smith and Sandwell topographic roughness h2

and rms tidal flows U used here. Some of the largest

internal tide sources are associated with abrupt topog-

raphy (Fig. 2) such as the Luzon Strait, Hawaiian Ridge,

Tuamotu Archipelago, Aleutian Island chain, and so on

(Ray and Mitchum 1997; Lee et al. 2006; Simmons et al.

FIG. 9. Depth–latitude sections of (a) number of data points n, (b) average buoyancy frequencyN, (c) dissipation rate «, and (d) diapycnal

diffusivityK for the (left) Indian, (center) Pacific, and (right) Atlantic. Dissipation rates « (c) are elevated in the pycnocline (latitudes, 508–608) mirroring the stratificationN in (b). DiffusivitiesK in (d) areO(0.13 1024) m2 s21 in much of the oceans but are elevated in the Indian

andAtlantic sectors of the SouthernOcean (latitudes below 408S) and in the northern NorthAtlantic (latitudes above 408N) and spottily near

the bottom. There are also hints of a band of elevated K just equatorward of the semidiurnal PSI critical latitude of 288 (Alford et al. 2007;

MacKinnon et al. 2013a,b) in the North and South Pacific but not the Atlantic or Indian. Black contours are density surfaces.

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2004; Zhao et al. 2016), for which the weak-topography

approximation does not apply. No effort was made to

isolate topographic scales relevant to internal tide and lee-

wave generation or to include fortnightly tidal modulation

or subinertial flows. We have no way of robustly isolating

tidal, lee-wave, and wind-forced dissipation sources. Nor

do we need internal wave sources and sinks to be corre-

lated in space/time because internal waves propagate (Ray

and Mitchum 1997; Alford 2001, Zhao et al. 2016), car-

rying energy away to dissipate elsewhere and at other

times. For example, the canonical q5 0.3 implies 70% of

the energy is not dissipated locally. We cannot distinguish

locally and remotely forced dissipations, both of which

Oka and Niwa (2013) find are necessary to explain the

Pacific thermohaline circulation.

Future research is planned to try to better tease apart

conditions needed for the turbulent bottom boundary

layer to extend through thewholewater column and howq

depends on topographic forcing, but, because of the limi-

tations in the data, process-oriented numerical modeling

may be the best way to explore this parameter space.

7. Conclusions

Afinescale parameterization for internal-wave-driven

turbulent dissipation rates « and diapycnal diffusivities

K was applied to ;30 000 CTD casts from all the major

oceans (Fig. 2), though excluding the Arctic, Weddell,

and Ross Seas. The global integrated dissipation of

2.0 6 0.6 TW is consistent with the 2.1 6 0.7 TW tide,

FIG. 10. Zonal averages of (a) vertically integrated dissipation ratesЫ, (b) topographic

roughness h2, and (c) rms tidal currents U as a function of latitude. Integrated dissipation

rates are for the water column below 380-m depth (black) in the pycnocline between 380- and

2000-m depth (red and below 2000-m depth (blue).

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wind, and lee-wave sources for internal gravity waves, so

there may be no need to invoke missing or ‘‘dark’’ tur-

bulent mixing on continental slopes and canyons (Kunze

et al. 2006; Waterhouse et al. 2014), though we caution

that uncertainties are large enough that, for example,

there need be little or no contribution from internal

lee waves (Waterman et al. 2014). Of this dissipation,

80%–90% occurs above 1000-m depth and less than

0.08 TW below 2000m (Fig. 8), compared to the 0.3 TW

required to maintain deep stratification in a vertical

advective–diffusive balance that ignores horizontal ad-

vection (Munk 1966; Munk and Wunsch 1998; Wunsch

and Ferrari 2004). As a caveat, because the bulk of

the dissipation occurs in the upper pycnocline, deep

(.2000-m depth) and abyssal (.4000-m depth) mixing

are poorly constrained by the global bulk budget.

Nevertheless, it can be concluded that most mixing is

remote from deep generation sites. The 256-m half-

overlapping spectral bins limit vertical resolution and

so may not resolve thin stratified turbulent bottom

boundary layers. Vertically integrated dissipation rates

Ы vary by three to four orders of magnitude (Fig. 2)

with elevated values in the western Indian and Pacific

Oceans associated with abrupt topography (Fig. 3),

consistent with internal tide generation site predictions

(Egbert and Ray 2001; Simmons et al. 2004; Nycander

2005). These do not scale with the predictions of linear

theories for internal tide or lee-wave generation (Bell

1975; Fig. 12), suggesting that the locally dissipated

fraction q decreases with increasing forcing, and sig-

nificant horizontal redistribution of wave energy occurs

before dissipation. Because there is little local dissipation/

mixing and considerable redistribution by internal wave

propagation, prediction of where and when turbulent

dissipation will occur is not straightforward.

Spatial patterns are repeatable (Figs. 4, 6) and show

variable turbulent bottom boundary layer thicknesses

that often extend throughout the whole water column

over rough topography (Figs. 4–5) in contrast to the

fixed 500-m decay scale assumed in OGCM tidal mixing

parameterizations (Simmons et al. 2004; Saenko and

Merryfield 2005; Jayne 2009; Friedrich et al. 2011). This

FIG. 11. Depth–longitude sections of turbulent diapycnal diffusivities K and buoyancy frequencies N binned by latitude. Diffu-

sivities areO(1025) m2 s21 in most of the ocean but become higher in the North Atlantic (608W–08) in the upper ocean of the western

boundaries and in parts of the Southern Ocean (308E –1808, 708–108W), where the stratification is weak. Elevated diffusivities in

the Southern Ocean correspond to the longitude bands where N is weak and there is elevated inertial wind forcing (Kilbourne 2015)

but are also at latitudes of the Antarctic Circumpolar Current where the finescale parameterization overestimates turbulence

(Waterman et al. 2014). Elevated diffusivities in the shallowest layer at high latitudes are likely biased high by sharp pynoclines so are

not to be trusted.

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supports use of a more dynamically motivated parame-

terization such as Polzin (2004) as implemented inMelet

et al. (2013), or Olbers and Eden (2013). Further testing

is needed to better determine how local dissipative

fraction q and decay scale depend on topography, tidal

flows, and other environmental properties.

The global-averaged turbulent diapycnal diffusiv-

ity K is almost independent of depth z at (0.3–0.4) 31024m2 s21 but increases from 1025m2 s21 at 6000mab to

1024m2 s21 at the bottom (h5 0) in height above bottom

h coordinates with an average e-folding scale of;2000m

(Fig. 7), though this decay scale is not constant (Fig. 4).

The difference between the z and h coordinate systems

arises because the bottom (h 5 0) is not always at the

same depth z. Diffusivities vary by two orders of

magnitude but cannot be estimated reliably for values

exceeding 10 3 1024 m2 s21 because of limitations in

the methodology. On average, the dissipation rate «

decreases with depth z and density (Fig. 7), though it

may display a weak increase with neutral densities

greater than 28.2. It is elevated to 1029Wkg21 in the

bottom 500 mab in height above bottom coordinates but

exhibits little variability over this bottom layer, the largest

gradient being between 700 and 1200 mab, and it is 0.1531029Wkg21 between 1000- and 3000-mab before in-

creasing slowly as h increases. This contrasts with the

commonly assumed exponential decay over 500 mab

above rough topography in OGCMs based on micro-

structure measurements in the Brazil Basin (St. Laurent

et al. 2001). This may reflect a difference between global

and regional behavior or that the finescale parameteri-

zation is underestimating near-bottom directly forced

turbulence. Consistent with the former interpretation,

an average profile that excludes the upper 2000m

(pycnocline) produces a turbulent bottom boundary

layer that more closely resembles the canonical decay

scale of;500 mab (dotted curves with h in Fig. 7). The

average dissipation rate is 0.2 3 1029Wkg21 for

FIG. 12. Full water column vertically integrated dissipation ratesЫ binned by (left) bottom roughness variance

,h2. over 30 km 3 30 km, (center) linear internal lee-wave generation [(7)], and (right) linear internal tide

generation [(6)]. The upper row displays the joint probability distributions and the lower row the means and

standard errors. All integrated dissipation ratesЫ exclude the upper 380m where strain estimates of « may be

contaminated by sharp changes in background stratification. Dotted curves in the lower row correspond to the

probability distributions of the lower-axis variable. Thick diagonal lines correspond to a linear dependence on the

horizontal axes (e.g., h2 in the left set of panels); thin lines indicate the corresponding square root and quartic root of

the same. Levels are not meaningful.

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buoyancy B , 20.268 (gn . 28.2), increasing to values

greater than 2 3 1029Wkg21 for B . 20.262 (gn ,27.0). Finescale inferences agree with average direct

microstructure measurements at the 17 sites high-

lighted in Waterhouse et al. (2014).

Zonal averages in all three oceans (Fig. 9) are similar,

with weak diffusivities along the equator despite elevated

strain variance (Fig. 6) because theN/f dependence in (4)

moderates the elevated strain; this contrasts with the off-

equatorial stripes of elevated dissipation rate reported in

the pycnocline by Whalen et al. (2012) based on more

extensive Argo profiling float sampling. Higher diffusiv-

ities are found at subpolar latitudes in the Indian and

Atlantic but not Pacific, reflecting both their stratification

and wind-forcing patterns. However, the finescale pa-

rameterization is also known to overestimate turbulence

in the Antarctic Circumpolar Current at these latitudes

(Waterman et al. 2014). The overall uniformity of K is

consistent with «;KN2, and most features in the zonally

averaged « can be related to variability in the stratifica-

tion rather than diffusivity. No compelling support for

tidal parametric subharmonic instability (PSI) enhancing

turbulence production equatorward of 148 and 288 lati-tudes was found (Figs. 9–10) since K is elevated near 308in the Pacific but not Atlantic or Indian. Likewise, in-

tegrated dissipation rates south of 408S are less than 0.03

TW (Figs. 2, 10), which does not support 0.1–0.3TW lee-

wave dissipation of Antarctic Circumpolar Currents

(Nikurashin and Ferrari 2011; Scott et al. 2011; Melet

et al. 2014; Wright et al. 2014) but is consistent with mi-

crostructure measurements finding dissipations as much as

an order of magnitude below theoretical predictions

(Waterman et al. 2014). Again, the finescale parameteri-

zation overestimates turbulent dissipation rates in Ant-

arctic Circumpolar Currents (Waterman et al. 2014).

The proxy dataset for global ocean mixing assembled

here has shown reasonable skill in reproducing direct

microstructure and preconceptions in a broad brush

view. While well suited for studying semisteady turbu-

lent sources such as internal tides, the datasetmay not be

suitable for studying wind-forced internal-wave-driven

mixing because of its limited temporal sampling; the

Argo profiling float dataset has proven more appropri-

ate for exploring these dependencies (Whalen et al.

2012) and finds little seasonal variability outside storm-

forced latitudes 308–408 (Whalen et al. 2015). The

finescale parameterization has also been found to

overestimate turbulence where internal lee waves are

expected to be the source (Waterman et al. 2014). How

well this dataset can reproduce the details of internal-

wave-driven mixing despite these known limitations

awaits further analysis and comparison, specifically

exploration of the decay scale and dissipative fraction q

of the stratified turbulent bottom boundary layer as-

sociated with internal tide generation in the context of

predictions based on internal wave–wave interaction

theory (Polzin 2004; Olbers and Eden 2013).

Acknowledgments. For Walter Munk, who started it

all, on his 100th birthday. The author acknowledges the

efforts of the hundreds of scientists and technicians who

collected, processed, and quality controlled the hydro-

graphic data used in this study. Barry Ma and Fiona Lo

assisted with data extraction. Discussions with Cimarron

FIG. A1. (left) Minimum frequency vmn (A3) and corresponding latitude (upper axis)

where vmn 5 f as functions of vertical wavenumber m (vertical coordinate). (right) Meridi-

onal length scale L 5 1/‘mn [(A2)] corresponding to the minimum frequency vmn [(A3)]

as a function of vertical wavenumber m. Low vertical wavenumbers are influenced by

latitudes;38, while finescale waves are much more closely confined within 20 km in their off-

equatorial influence.

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Wortham on fitting procedures were valuable. Com-

ments from two anonymous reviewers led to improvements

of the manuscript. This research was supported by

NSF Grant OCE-1153692. (Internal-wave-driven infer-

red turbulence dataset is available at ftp.nwra.com/

outgoing/kunze/iwturb.)

APPENDIX

Equatorial Internal Waves

Internal gravity waves on the equator are trapped

inside a meridional waveguide by b 5 ›f/›y such that

they feel off-equatorial rotation f 5 by by virtue of

having finite meridional scales. Therefore, the equato-

rial internal wave dispersion relationmight be written as

v5b

‘1

N2(k2 1 ‘2)‘

2bm2, (A1)

where ‘ is proportional to an effective meridional

wavenumber, k is the zonal wavenumber, and m is the

vertical wavenumber. Equation (A1) has a minimum

frequency at k 5 0 and ‘, satisfying

052b

‘21

3N2‘2

2bm20 ‘

mn5

�2

3

�1/4�bm

N

�1/2(A2)

with corresponding frequency

vmn

5

"�3

2

�1/41

1

2

�2

3

�3/4#�

bN

m

�1/2(A3)

(Fig. A1). If this scaling is correct, it may be more

appropriate to use vmn [(A3)] instead of f in the

N/f scaling [(4)] near the equator to allow finite tur-

bulence production, though Fig. A2 illustrates that

wavelengthsO(10)m are meridionally confined within

;0.48 of the equator and so will have very low vmn ;1026 rad s21 (;2-month periods). In contrast, low ver-

tical modes with O(1000)m vertical wavelengths are

confined within;28 latitude withvmn; 53 1026 rad s21

(;2-week periods).

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